On the distribution of $ϕ(ψ(n))$ and $ψ(ψ(n))$

[2608.03749] On the distribution of $ϕ(ψ(n))$ and $ψ(ψ(n))$ Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2608.03749 (math) This paper has been withdrawn by Aimin Guo [Submitted on 4 Aug 2026 ( v1 ), last revised 24 Aug 2026 (this version, v3)] Title: On the distribution of $ϕ(ψ(n))$ and $ψ(ψ(n))$ Authors: Aimin Guo No PDF available, click to view other formats Abstract: Let $\psi(n)$ and $\phi(n)$ denote Dedekind's arithmetic function and Euler's totient function, respectively. We study the distribution of the compositions $\phi(\psi(n))$ and $\psi(\psi(n))$. In particular, we obtain quantitative upper bounds for the exceptional set associated with $\phi(\psi(n))$, thereby refining a density result of Sándor. We also prove that, for every fixed $c>0$, the set of integers $n\leq x$ satisfying $\psi(\psi(n))\leq cn$ has asymptotic density zero. Our method adapts sieve ideas used by Dixit and Bhattacharjee to compositions involving Dedekind's arithmetic function. Comments: The paper requires substantial revisions and updates. We are withdrawing it to prepare a significantly improved replacement version Subjects: Number Theory (math.NT) MSC classes: 11A Cite as: arXiv:2608.03749 [math.NT] (or arXiv:2608.03749v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.03749 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Aimin Guo [ view email ] [v1] Tue, 4 Aug 2026 14:40:00 UTC (7 KB) [v2] Thu, 6 Aug 2026 09:14:42 UTC (7 KB) [v3] Mon, 24 Aug 2026 22:24:45 UTC (1 KB) (withdrawn) Full-text links: Access Paper: Withdrawn Current browse context: math.NT < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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